Structure-Conveying Modelling Language SML User's Guide
Contents
1. 000000000000 0000 15 4 2 Implementation 2 2 200 ad a a eS 16 Bibliography 18 Introduction Foreword Algebraic modelling languages are recognised as an important tool in the formulation of mathematical programming problems They facilitate the construction of models through a language that resembles mathematical notation and offer convenient features such as automatic differentiation and direct inter facing to solvers Their use vastly reduces the need for tedious and error prone coding work Examples of popular modelling languages are AIMMS 1 GAMS 2 Xpress Mosel 3 and AMPL 4 SML Structured Modelling Language attempts build on these traditional approaches while encapsu lating the structure of the problem As such it encapsulates the AMPL syntax in an object oriented fashion We believe that object oriented modelling will eventually confer the following benefits Clarity of modelling reducing the apparent complexity of models Reusability of model parts reducing errors and allowing more complex systems to be modelled Planned for later releases Speed of solution encouraging a structured model which can be exploited by transparently passing structural information to the solver References It would be greatly appreciated if users would cite the following references in work that uses SML e Marco COLOMBO ANDREAS GROTHEY JONATHAN HOGG KRISTIAN WOODSEND AND JACEK GONDZIO A Structure Conveying Modell2. 3 1 Blocks A block is defined as follows block nameofblock 7 in set_ezpr statements Between the curly braces any number of set param subject to var minimize or indeed further block definitions can be placed The square brackets are not part of the language and are merely used to indicate optinal parts of the expression The interpretation is that all declarations placed inside the block environment are implicitly repeated over the indexing expression used for the block command Clearly the nesting of such blocks creates a tree structure of blocks A block introduces the notion of scope similar to that found in the programming language C Within a block all references to objects visible to the containing context are visible unless there is an object with the same name defined within the block 3 2 Stochastic Blocks and Variables A stochastic program is declared in SML via the stochastic modifier to the block command block nameofblock k in set_expr stochastic using i in NODES PROB PARENT j in STAGESET stochastic statements 11 where the expressions NODES PROB PARENT and STAGESET conform to the following relations set NODES param PROB NODES param PARENT NODES symbolic within NODES union none set STAGESET ordered The stochastic statements include both normal AMPL and block statements but may also include the special stages statement stages set_expr stochas
3. if comm_source k i then comm_demand k else if comm_target k i then comm_demand k else 0 var sparecap ARCS gt 0 block MCNFArcs a in ARCS set ARCSDIFF ARCS diff a local ARCS still present block Net k in COMM var Flow ARCSDIFF gt 0 flow into node flow out of node equals demand subject to FlowBalance i in NODES sum j in ARCSDIFF arc_target j i Flow j sum j in ARCSDIFF arc_source j i Flow j b k i subject to Capacity j in ARCSDIFF sum k in COMM Net k Flow j lt basecap j sparecap j block MCNFNodes n in Nodes set NODESDIFF NODES diff n local NODES still present set ARCSDIFF i in ARCS arc_source i n arc_target i n block Net k in COMM var Flow ARCS gt 0 flow into node flow out of node equals demand subject to FlowBalance i in NODESDIFF sum j in ARCSDIFF arc_target j i Flow j sum j in ARCSDIFF arc_source j i Flow j b k il subject to Capacity j in ARCSDIFF sum k in COMM Net k Flow j lt basecap j sparecap j minimize costToInstall sum j in ARCS sparecap j cost j 2 4 Stochastic Structure As stochastic programming is a common source of such structured problems and the special conventions used to represent stochastic entities SML includes some special conventions for handling stochastic problems Our next example shows these in action 2 4 1 ALM Example param Budget
4. Bibliography 1 J BISSCHOP AND R ENTRIKEN AIMMS The Modeling System Paragon Decision Technology 1993 2 A BROOKE D KENDRICK AND A MEERAUS GAMS A User s Guide The Scientific Press Redwood City California 1992 3 Y COLOMBANI AND S HEIPCKE Mosel an extensible environment for modeling and program ming solutions in Proceedings of the Fourth International Workshop on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimisation Problems CP AI OR 02 N Jussien and F Laburthe eds Le Croisic France Mar 25 27 2002 pp 277 290 4 R FOURER D Gay AND B W KERNIGHAN AMPL A Modeling Language for Mathematical Programming The Scientific Press San Francisco California 1993 5 J GONDZIO AND R SARKISSIAN Parallel interior point solver for structured linear programs Math ematical Programming 96 2003 pp 561 584 18
5. T Tc Rho set ASSETS NODES STAGESET 0 T param PARENT NODES symbolic PROBS NODES param Value ASSETS var mu block alm stochastic using NODES PARENT PROBS STAGESET var xh ASSETS gt 0 xb ASSETS gt 0 xs ASSETS gt 0 param Ret ASSETS param Liability deterministic stage 0 subject to StartBudget 1 Tc sum j in ASSETS xb j Value j lt Budget stage 1 T subject to Inventory j in ASSETS xh j 1 Ret j ancestor 1 xh j xb j xs jl subject to CashBalance 1 Tc sum j in ASSETS Value j xs j Liability 1 tc sum j in ASSETS Value j xb j stage T var wealth 1 tc sum j in ASSETS Value j xh j subject to ExpPortfolioValue Exp wealth mu maximize objFunc mu Rho wealth wealth mu mu 10 3 SML Keywords In order to facilitate Structured Modelling we have introduced several new keywords and structures into the AMPL language These keywords are the following reserved words which may not be used for component names ancestor Exp stochastic block node suffix deterministic stages using We also note that while AMPL allows the redefinition of certain keywords for example from and prod rendering their erstwhile meanings unavailable sml does not permit this for the following reasons 1 It complicates the coding required to handle them 2 We believe it makes the model more error prone and hard to read
6. be expressed by using the ancestor i function that allows to reference the i th ancestor stage Variables of the ancestor can be referenced using the normal SML syntax ancestor i x Expectations may also be handled through a special new syntax Within a stochastic block the expression Exp expr is equivalent to the expression sum nd in NODES nd in currentstage PROBS nd expr nd where currentstage is a list of nodes in the scope of the current stages block 12 4 Advanced Topics 4 1 Solver Interface If you wish to hook a solver other than OOPS to SML then you will need to use the interface described in this section We first describe the data representation used by the interface before describing the available function calls The implementer is advised to browse the source code for the MPS interface for an example implementation 4 1 1 Data structure Internally SML describes a problem as a nested bordered block diagonal form The data structure used to represent this assumes the following 1 A block has a zero or more constraints of variables with an associated Jacobian matrix labelled A in Figure 4 1 2 Further associated variables may contribute to the objective function 3 A block has zero or more additional blocks nested within it these are represented by a list of pointers to the contained blocks 4 A contained block has a similar internal structure labelled B that has no bearing on the constraints of
7. expression of the sub block and this is prepended to the indexing expressions of every entity declared in the sub block By temporarily defining the set ARCS_SUB to a suitable subset of ARCS leveraging AMPL s scripting commands the model for any node on the expanded tree can be generated from the corresponding submodel mod file shown in Figure 4 2 This process of model expansion based on indexing sets results in a n1 file for every node of the expanded model tree each file carries all the information about this node needed by a solver The underlying submodel is the same for all expanded tree nodes that correspond to the same prototype tree node However they are produced with different data instances namely different choices of elements from the block s indexing sets The nodes of the prototype and the expanded tree are internally represented as C objects that carry pointers to their children Therefore the prototype and expanded trees are themselves trees of C objects The ExpandedTreeNode class provides information on the dimension of the node number of local constraints and variables and a list of its children further it provides methods to evaluate the Jacobian of its local constraints with respect to the local variables of a second ExpandedTreeNode object Information on the number of sparse elements of this Jacobian block can be obtained prior to requesting the complete sparse matrix description to enable the allocation of su
8. i 7 i is its parent in the tree p is the probability of reaching it and 7 i represents its stage With Lr we denote the set of final stage nodes The objective maximises a linear combination of the expectation of the final portfolio value denoted by u and its variance with risk aversion parameter A max JE wealth Var wealth max JE wealth A wealth E wealth The complete model is the following max p pil 1 c Y gt ja y x w20 iELT jEA Maca tia la HD je cjujt y Vi 0 jea Mj 0 5 b LSC apa GEA Up ST 2 3 Identifying the structure How do we identify the structure Generally if you are building you model out of repeated components you can easily obtain structure it may be that this structure would be indexed over in the traditional Algebraic Modelling Language representation for example the indexing over ARCS and NODES in the MSND example and the indexing over the scenario tree in the stochastic ALM example If this indexing is pulled out into a series of related blocks then the problem becomes similar In the next example we consider the structured representation of the MSND model 2 3 1 MSND Example Using the block keyword the previous MSND model can be rewritten as set NODES ARCS COMM param cost ARCS basecap ARCS param arc_source ARCS arc_target ARCS param comm_source COMM comm_target COMM comm_demand COMM param b k in COMM i in NODES
9. ModelInterface object The solver should use the include statement include sml h which will provide access to all relevant definitions The solver should then call the function ExpandedModelInterface sml_generate const std string modelfilename const std string datafilename const bool debug to retrieve a ExpandedModel Interface object which can then be queried If the argument debug is true then additional debugging information will be generated Traversing the Tree The ExpandedModelInterface object returned after parsing the model and data files offers two iterators for moving through the tree and further allows direct access to the list of children e The ExpandedModelInterface ancestor_iterator allows the iteration up through the tree to the root The function ExpandedModelInterface abegin returns an iterator which points to the parent of the node it is called on The function ExpandedModel Interface aend return an iterator which represents the top of the tree above the root 14 e The ExpandedModel Interface child_iterator allows the iteration over all children of a node in a depth first order with the node itself being last The function ExpandedModelInterface cbegin returns an iterator pointing to the left most leaf in the subtree routed at the calling node The function ExpandedModelInterface cend returns an iterator representing the end of the child iteration sequence e The component std vector l
10. Structure Conveying Modelling Language SML User s Guide Marco Colombo Andreas Grothey Jonathan Hogg Kristian Woodsend Jacek Gondzio School of Mathematics and Maxwell Institute University of Edinburgh UK http www maths ed ac uk ERGO sm1 sml support maths ed ac uk February 18 2011 Table of Contents Introduction 3 1 Installation 4 1 1 Third party software a od e COA EERE BOY OL Se RSS eee a 4 1 2 Configuration Options 3 4 areor A Be A ee ee E a ee ee 5 1 37 Rumning SMG ae as ee ares SG ane aed Ges Sw GL Get Ae ewe ale a Gee ce 5 2 Structured Problems 6 2 1 Example Problem 1 Survivable Network Design o e 6 2 1 1 Standard modelling language formulation 7 2 2 Example Problem 2 Asset and Liability Management 8 2 3 Identifying the structure 2 ta a a Sate E a ee Ge Ba 9 2 3 1 MSND Example agr ne wa 64 A ee ee ee ee eS 9 2 4 Stochastic Structtires alee e ee RE a a ee heh A 10 2 41 ALM Ezamplei ss dee e ee A ee Rc em aa 10 3 SML Keywords 11 del Blocks a totes eg bole a Og E aa es Mae ER A ay 11 3 2 Stochastic Blocks and Variables o 11 4 Advanced Topics 13 AV Solver Interface os cost uta Bae hal e Bae dt bile Gn SS 13 ALA Data structure a Gea Bw a aod Ae Oe a de be th ce 13 AN 2 Basie USE gi A eb Gh Aa he ae ROR BA E AAA 13 AE APE a anes Sete a RY De A ey et i Re a oa R 14 4 1 4 Returning the solution
11. cient to build and use SML through the smlmps interface Other two interfaces smloops and smlcplex can be built if additional software libraries are available The third party software you will need is summarised in Table 1 1 Software Required Free URL AMPL Yes No http www ampl com vendors html amplsolver a Yes Yes http www netlib org ampl solvers OOPS No Limited http www maths ed ac uk gondzio parallel solver html BLAS for OOPS Yes http www netlib org blas LAPACK for OOPS Yes http www netlib org lapack METIS No Yes http glaros dtc umn edu gkhome views metis Cplex No No http ilog com products cplex Table 1 1 Third party software and where to get it OOPS is the Object Oriented Parallel Solver by Gondzio and Grothey 5 If you wish to use the OOPS solver then you will also need the OOPS library A serial version limited to 5000 constraints is distributed with SML under a closed source license and may only be linked with the LGPL SML library See ThirdParty oops COPYING for full terms The version supplied is compiled using GNU g and gfortran on a 32 bit Linux machine if your setup differs then mileage may vary If you have a full version of OOPS contact Andreas Grothey lt A Grothey ed ac uk gt for further infor mation you may use this instead though you may also need the METIS library OOPS also requires LAPACK and BLAS routines we advise that you use the ones supplied by your computer vendo
12. ed then this can be exploited by modern solvers to solve the problem faster This structure can take a number of forms for example if the problem involves a network then the node arc incidence matrix may be repeated multiple times within the matrix A or if the problem is stochastic in nature then the scenario tree can be exploited Throughout this guide we will consider two structured problems MSND Survivable Network Design ALM Asset and Liability Management 2 1 Example Problem 1 Survivable Network Design A network is described by sets V of nodes and of directed arcs and a base capacity Cj for every arc j The basic multi commodity network flow MCNF problem considers the best way to move commodities k C from their sources to their respective destinations through the arcs of the shared capacity network Each commodity is described as the triplet sx tk dp consisting of start node sg terminal node t and amount to be shipped dp A feasible flow x k j jee for the k th commodity can be represented by the constraint AZXk bk Tk gt 0 where A IRIYIXIEI is the node are incidence matrix and bx bk i icy is the demand vector for the k th commodity with the following conventions 1 node 7 is source of arc j dr node 7 is source of demand k Ay 1 node 1 is target of arc j Des dk node 2 is target of demand k 0 otherwise 0 otherwise In multi commodity survivable network design MSND the a
13. f the constraint matrix for the MSND problem The constraint matrix of this problem has the form shown in Figure 2 1 The basic building blocks are the node arc incidence matrices shown in dark grey These blocks are repeated for every commodity that needs to be shipped and they are linked by a series of common rows shown in medium grey that represent the global capacity constraints to build a MCNF problem Each MCNF block framed by dashed lines is repeated for every missing arc and node The common capacity variables light grey blocks act as linking columns While the nested structure of the problem is obvious is Figure 2 1 it cannot be easily appreciated from the mathematical formulation 2 2 2 1 1 Standard modelling language formulation Problem 2 2 can be represented in an algebraic modelling language as shown below where we adopt a syntax that is very close to AMPL slightly modified to compress the example and aid readability models in other languages will look much the same set NODES ARCS COMM param cost ARCS basecap ARCS param arc_source ARCS arc_target ARCS param comm_source COMM comm_target COMM comm_demand COMM param b k in COMM i in NODES if comm_source k i then comm_demand k else if comm_target k i then comm_demand k else 0 first index is missing arc node then commodity then arc of flow var Flow ARCS union NODES COMM ARCS gt 0 var sparecap ARCS gt 0 flow i
14. fficient memory As argued above these methods should satisfy the needs of all different conceivable structure exploiting solvers 16 set ARCS param cost ARCs var sparecap ARCS gt 0 minimize obj sum j in ARCS sparecap j cost j set ARCS COMM param basecap ARCS var sparecap ARCS gt 0 set ARCS_SUB within ARCS set ARCSDIFF a in ARCS_SUB ARCS diff a local ARCS still present var MCNFArcs_Net_Flow a in ARCS_SUB ARCSDIFF a k in COMM gt 0 subject to Capacity a in ARCS_SUB j in ARCSDIFF a sum k in COMM MCNFArcs_Net_Flow a k j lt basecap j sparecap j set NODES ARCS COMM param arc_source ARCS arc_target ARCS param comm_source COMM comm_target COMM comm_demand COMM param b k in COMM i in NODES if comm_source k i then comm_demand k else if comm_target k i then comm_demand k else 0 set ARCS_SUB within ARCS set ARCSDIFF a in ARCS ARCS diff a local ARCS still present set COMM_SUB within COMM var MCNFArcs_Net_Flow a in ARCS_SUB k in COMM_SUB ARCSDIFF a gt 0 flow into node flow out of node equals demand subject to MCNFArcs_Net_FlowBalance a in ARCS_SUB k in COMM_SUB i in NODES sum j in ARCSDIFF a arc_target j i MCNFArcs_Net_Flowl a k j sum j in ARCSDIFF al arc_source j i MCNFArcs_Net_Flow a k j b k il Figure 4 2 Generated AMPL model files for the root MSND model and MCNF submodels 17
15. he complete solution may be output recursively for a block ie the solution associated with this block and all its descendants by calling the function void outputSolution std ostream amp out int indent which will write the solution to the output stream out If the optional argument indent is supplied that the solution will be indented by a number of spaces equal to indent If indent is not present then no indent will be used 4 2 Implementation SML is implemented in the object oriented C language as a pre and post processor to AMPL The SML model file is parsed to extract the prototype model tree the list of entities associated with each prototype tree node and the dependency graph of the entities The goal of the pre processing stage is to create for each node in the prototype model tree a stand alone AMPL model file that describes the local model for this block of the problem The file includes definitions for all the entities belonging to the prototype tree node and all their dependencies references to entities are changed to a global naming and indexing scheme that easily generated from the SML model file by generic text substitutions Figure 4 2 shows these AMPL submodel files for the MSND problem formulation The AMPL model is separated into the appropriate submodels for every node of the prototype tree by changing the indexing sets Each block definition is replaced by declaring an indexing set named _SUB for the indexing
16. im is to find the minimum installation cost of additional spare capacities S at price cj 7 so that the given commodities can still be routed through the network if any one arc or node should fail The MSND problem can be modelled by a series of multi commodity network flow problems in each of which one of the original arcs or nodes is removed Note that the subproblems are not completely independent as they are linked by the common spare capacities S Let A 1 E be the node arc incidence matrix in which the column corresponding to the l th arc is set to zero Then any vector of flows ol gt 0 satisfying ACD go bx kEC IEE gives a feasible flow to route the k th commodity through the arc reduced network Similarly let Alri i V be the node arc incidence matrix in which the row corresponding to the i th nodes and the columns corresponding to arcs incident to this node are set to zero Then any vector of flows al gt 0 satisfying AMG bp KECLEY gives a feasible flow to route the k th commodity through the node reduced network As the network is capacity limited however each arc j can carry at most Cj S units of flow The complete formulation of the MSND problem is as follows min gt C55 jeE st Al gl by VkeC le ACDD by VEEC iEV Sia lt C 8 VjeE lee 2 keC Saf C 8 VYjEeE eV keC r s gt 0 porro nnanansl encon Figure 2 1 Structure o
17. ing Language for Mathematical and Stochastic Program ming Technical Report ERGO 09 003 School of Mathematics Accepted for publication in Math ematical Programming Computation ANDREAS GROTHEY JONATHAN HOGG KRISTIAN WOODSEND MARCO COLOMBO AND JACEK GONDZIO A Structure Conveying Parallelizable Modelling Language for Mathematical Program ming Springer Optimization and Its Applications Vol 27 Parallel Scientific Computing and Op timization ed R Ciegis D Henty B Kagstr m and J Zilinska 2009 Reproduction This user s guide is licensed as part of SML under the Lesser GNU Public License LGPL version 3 You can redistribute it and or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation using version 3 of the License You should have received a copy of the GNU Lesser General Public License along with this program If not see http www gnu org licenses 1 Installation SML uses the standard GNU autotools install process This can be summarised by the following steps 1 configure with ampl path to amplsolver 2 make 3 make install However there are number of options and pieces of third party software you will require 1 1 Third party software As SML is currently implemented as a pre and post processor to AMPL you will need an installed version of AMPL You will also need the Amplsolver library available from Netlib These two components are suffi
18. jective coefficients for block The Compressed Sparse Column CSC format used by getJacobian0f Intersection describes for each column only the non zero entries within it and for each entry gives a row number and value pair On entry the parameters should point to areas of memory corresponding to the following arrays int colbeg n 1 int collen n int rownbs nz double el nz where n is the number of variables in this model and nz is the number returned by the corresponding call to getNzJacobian0f Intersection On return colbeg i is a pointer to the beginning of column i within the arrays rownbs and el collen i is the number of entries in that column which may be zero The values rownbs j and el j represent a row number value pair 4 1 4 Returning the solution You may return the solution to SML and have it write out a file containing these solutions The following functions are available for each ExpandedModelInterface block to return the primal and dual solutions associated with each variable and constraint within the block void setPrimalSolColumns double elts Set the primal solution vector for variables in this block void setDualSolColumns double elts Set the dual solution vector for variables in this block void setPrimalSolRows double elts Set the primal solution vector for constraints in this block void setDualSolRows double elts Set the dual solution vector for constraints in this block 15 T
19. nto node flow out of node equals demand subject to FlowBalanceMissingArcs a in ARCS k in COMM i in NODES sum j in ARCS j7 a arc_target j i Flow a k j sum j in ARCS j a arc_source j i Flow a k j b x il subject to FlowBalanceMissingNodes n in NODES k in COMM i in NODES diff n sum j in ARCS arc_target j i arc_source j n Flow n k j sum j in ARCS arc_source j i arc_target j n Flow n k j b x i subject to CapacityMissingArcs a in ARCS union NODES j in ARCS sum k in COMM Flow a k j lt basecap j sparecap j minimize obj sum j in ARCS sparecap j cost j 2 2 Example Problem 2 Asset and Liability Management As an example of a stochastic programming problem we use an asset and liability management problem The problem is concerned with investing an initial cash amount b into a given set of assets A over T time periods in such a way that at every time stage 0 lt t lt T a liability payment l can be covered Each asset has value vj 7 A The composition of the portfolio can be changed throughout the investment horizon incurring proportional transaction costs c a j 18 the amount of asset j held at node 2 a and x j are the amounts bought and sold These satisfy the inventory and cash balance equations at every node The random parameters are the asset returns r j that hold for asset j at node 7 The evolution of uncertainties can be described by an event tree For a node
20. r however there are freely available implementations available from the Netlib archive 1 2 Configuration Options The following configuration options are available with ampl path Absolute path to directory containing libamplsolver a with oops path Absolute path to OOPSHOME directory with cplex path Absolute path to the Cplex installation with blas library BLAS library to use with lapack library LAPACK library to use with metis path Absolute path to METIS directory with mps yes no Enable disable MPS interface Running the command configure help will display a more comprehensive summary 1 3 Running SML You can find some small examples of problems modelled in SML in the tests directory There are two files for each problem the model file and the data file This forces the user to write a model that is general enough so that it can be reused with different data sets The smlmps interface parses the model and data files to produce an MPS file which can then be used as input file to virtually any solver for solution For example this can be accomplished with the following command smlmps alm mod alm dat alm mps e To see how an interface can be called and what options it supports call it with the help option such as smlmps help 2 Structured Problems A typical mathematical programming problem can be written in the form min f x s t Ar b x gt 0 2 1 If the matrix A is structur
21. t ExpandedModelInterface gt ExpandedModelInterface children allows the implementer to use their own iteration scheme if the above are not sufficient We note that non equivalence of pointers is not a sufficient condition to differentiate nodes and the function ExpandedModelInterface getName should be used instead to compare the names of the blocks which are guaranteed to be unique Accessing the Data Each block essentially allows the query of all data in its row block The following functions are available int getNLocalVars Number of variables local to block const std list lt std string gt amp getLocalVarNames List of variable names local to block int getNLocalCons Number of constraints local to block const std list lt std string gt amp getLocalConNames List of constraint names local to block int getNzJacobian0fIntersection ExpandedModelInterface emcol Number of non zeros in submatrix in block column emcol and this block row void getJacobian0fIntersection ExpandedModelInterface emcol colbeg collen rownbs el Entries of submatrix in block column emcol and this block row in CSC format see below void getRowLowBounds double elts Constraint lower bounds for block void getRowUpBounds double elts Constraint upper bounds for block void getColLowBounds double elts Variable lower bounds for block void getColUpBounds double elts Variable upper bounds for block void getObjGradient double elts Ob
22. tic statements that declares variables and constraints which apply only to the specific stages in the supplied set expres sion Currently no check is performed on the validity of the values indicated in the PROB set the user has to be careful to indicate positive values such that for all stages their sum is 1 The scenario tree is defined by the set of NODES and their PARENT relationship The probability associated with a node is the conditional probability of reaching that node given that its parent is reached The set expression STAGESET is used only to provide labels for the stages should they be required It is assumed that the number of levels thus defined in the tree matches the cardinality of tht STAGESET set Entities defined within the stochastic block are either stochastic and are repeated for nodes in their stageset or are deterministic and are repeated only once for each stage By default entities are stochastic and their stageset is the full set STAGESET Entities within a stages block are by default stochastic with their stageset determined by the stages set expression The nature of an entity may be changed from the default through the use of the special attributes deterministic and stages set_expr in their declaration The dummy indices of the NODES and STAGESET set expressions allow the reference of the particular node or stage within the block through the normal indexing mechanism Relations between different stages can
23. variables associated with this block 5 The constrained block may contain constraints on the variables associated with this block with a Jacobian labelled C 6 This block may place constraints upon variables associated with the contained block yielding the Jacobian labelled D 7 Any or all of the Jacobians A B C or D may be identically zero The list of pointers to contained blocks yields a tree structure which we refer to as the Expanded Model Tree This representation of the structured model is natural for a hierarchical linear algebra solver such as OOPS but should also provide sufficient access for other designs of structure exploiting solver such as those which use decomposition approaches 4 1 2 Basic Use The model currently used by SML is that it is called by the solver and passed the names of the model and data files it will then return a pointer to an Expanded Model Tree represented by a ExpandedModelInterface object which the solver is then free to query Finally the solver should delete the ExpandedModelInterface object to free the memory 13 Figure 4 1 Structure of the constraint matrix for the MSND problem We recommend that solvers parse the tree in either one or two passes The initial parse may extract information on the dimension and relative density of the tree setting up data structures are required while the second pass will determine the actual numeric values 4 1 3 API Acquiring the Expanded
Download Pdf Manuals
Related Search